Finding the original amount after a percent change

Reverse direction. You are given the result and asked for what it started as. Divide by the multiplier instead of multiplying by it.

01 Worked Example Video

Every step of Finding the original amount after a percent change explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

After a decrease

A company’s workforce decreased by 35%. There are now 4,550 employees.

How many were there before? Type just the number.

A bigger drop

Attendance at a festival fell 40% this year, to 7,320 people.

How many attended last year? Type just the number.

After an increase

Enrolment rose 15% this year, to 8,050 students.

How many were enrolled last year? Type just the number.

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

forwards: \(\text{new} = \text{original} \times \text{multiplier}\) the shape you already know

backwards: \(\text{original} = \dfrac{\text{new}}{\text{multiplier}}\) same multiplier, divided instead

Work out the multiplier first and the rest is one division. A 26% fall means 74% survives, so the multiplier is 0.74. A 15% rise means the multiplier is 1.15. Then divide the number you were given by it.

Common slips

(1) Added the percent back on. If something fell 26%, adding 26% of the new figure does not return you to the original, because 26% of a smaller number is a smaller amount. (2) Divided by the wrong decimal. A 26% decrease divides by 0.74, not 0.26. (3) Missed “is 74% of” versus “74% off”. The first divides by 0.74, the second by 0.26. Read it twice.

04 Practice — Step by Step

Multiplier, then divide

A town’s population fell by 18% over a decade. It is now 9,840.

What was it before?

Step 1

What fraction survived?

If 18% was lost, what percent remains?

Step 2

Divide

9,840 ÷ 0.82. Type just the number.

Step 3

Check it forwards

12,000 × 0.82 — does it return you to 9,840?

Q3 Q5

Finding the original amount after a percent change

Why adding it back fails